{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:44:03Z","timestamp":1760240643278,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2019,7,29]],"date-time":"2019-07-29T00:00:00Z","timestamp":1564358400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11361008"],"award-info":[{"award-number":["11361008"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>For positive integers s and t, the Ramsey number     R ( s , t )     is the smallest positive integer n such that every graph of order n contains either a clique of order s or an independent set of order t. The triangle-free process begins with an empty graph of order n, and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. It has been an important tool in studying the asymptotic lower bound for     R ( 3 , t )    . Cyclic graphs are vertex-transitive. The symmetry of cyclic graphs makes it easier to compute their independent numbers than related general graphs. In this paper, the cyclic triangle-free process is studied. The sizes of the parameter sets and the independence numbers of the graphs obtained by the cyclic triangle-free process are studied. Lower bounds on     R ( 3 , t )     for small t\u2019s are computed, and     R ( 3 , 35 ) \u2265 237    ,     R ( 3 , 36 ) \u2265 244    ,     R ( 3 , 37 ) \u2265 255    ,     R ( 3 , 38 ) \u2265 267    , etc. are obtained based on the graphs obtained by the cyclic triangle-free process. Finally, some problems on the cyclic triangle-free process and     R ( 3 , t )     are proposed.<\/jats:p>","DOI":"10.3390\/sym11080955","type":"journal-article","created":{"date-parts":[[2019,7,29]],"date-time":"2019-07-29T03:06:58Z","timestamp":1564369618000},"page":"955","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The Cyclic Triangle-Free Process"],"prefix":"10.3390","volume":"11","author":[{"given":"Yu","family":"Jiang","sequence":"first","affiliation":[{"name":"College of Electronics and Information Engineering, Beibu Gulf University, Qinzhou 535011, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Meilian","family":"Liang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Information Science, Guangxi University, Nanning 530004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yanmei","family":"Teng","sequence":"additional","affiliation":[{"name":"School of Mathematics and System Science, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaodong","family":"Xu","sequence":"additional","affiliation":[{"name":"Guangxi Academy of Sciences, Nanning 530007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,7,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1112\/plms\/s2-30.1.264","article-title":"On a problem of formal logic","volume":"s2\u2013s30","author":"Ramsey","year":"1930","journal-title":"Proc. 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